\(\int \sqrt {a+b x} (c+d x)^{3/2} \, dx\) [1473]

Optimal result

Integrand size = 19, antiderivative size = 151 \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}} \]

[Out]

1/3*(b*x+a)^(3/2)*(d*x+c)^(3/2)/b-1/8*(-a*d+b*c)^3*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(5/2
)/d^(3/2)+1/4*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^2+1/8*(-a*d+b*c)^2*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {52, 65, 223, 212} \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=-\frac {(b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 b^2 d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b} \]

[In]

Int[Sqrt[a + b*x]*(c + d*x)^(3/2),x]

[Out]

((b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^2*d) + ((b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(4*b^2) +
((a + b*x)^(3/2)*(c + d*x)^(3/2))/(3*b) - ((b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x
])])/(8*b^(5/2)*d^(3/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}+\frac {(b c-a d) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{2 b} \\ & = \frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}+\frac {(b c-a d)^2 \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b^2} \\ & = \frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^2 d} \\ & = \frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^3 d} \\ & = \frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 b^3 d} \\ & = \frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84 \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^2 d^2+2 a b d (4 c+d x)+b^2 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 b^2 d}-\frac {(b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{5/2} d^{3/2}} \]

[In]

Integrate[Sqrt[a + b*x]*(c + d*x)^(3/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*a^2*d^2 + 2*a*b*d*(4*c + d*x) + b^2*(3*c^2 + 14*c*d*x + 8*d^2*x^2)))/(24*b^2*
d) - ((b*c - a*d)^3*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(8*b^(5/2)*d^(3/2))

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{2}}}{3 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 d}\)	\(173\)

[In]

int((b*x+a)^(1/2)*(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/3/d*(b*x+a)^(1/2)*(d*x+c)^(5/2)-1/6*(-a*d+b*c)/d*(1/2*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b-3/4*(a*d-b*c)/b*((b*x+a)
^(1/2)*(d*x+c)^(1/2)/b-1/2*(a*d-b*c)/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/2*b*c
+b*d*x)/(b*d)^(1/2)+(b*d*x^2+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.72 \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=\left [-\frac {3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{2} + 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \, {\left (7 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{3} d^{2}}, \frac {3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{2} + 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \, {\left (7 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{3} d^{2}}\right ] \]

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*
d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(8*
b^3*d^3*x^2 + 3*b^3*c^2*d + 8*a*b^2*c*d^2 - 3*a^2*b*d^3 + 2*(7*b^3*c*d^2 + a*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*
x + c))/(b^3*d^2), 1/48*(3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b*d)*arctan(1/2*(2*b*d*x
+ b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(8*b^
3*d^3*x^2 + 3*b^3*c^2*d + 8*a*b^2*c*d^2 - 3*a^2*b*d^3 + 2*(7*b^3*c*d^2 + a*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x
+ c))/(b^3*d^2)]

Sympy [F]

\[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=\int \sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((b*x+a)**(1/2)*(d*x+c)**(3/2),x)

[Out]

Integral(sqrt(a + b*x)*(c + d*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (119) = 238\).

Time = 0.40 (sec) , antiderivative size = 576, normalized size of antiderivative = 3.81 \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=-\frac {\frac {24 \, {\left (\frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} a c {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}}\right )} d {\left | b \right |}}{b} - \frac {6 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} c {\left | b \right |}}{b^{2}} - \frac {6 \, {\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} a d {\left | b \right |}}{b^{3}}}{24 \, b} \]

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-1/24*(24*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/sqrt(b*d)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a))*a*c*abs(b)/b^2 - (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqr
t(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*
d^3 - 11*a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*log(abs(-sqrt(b*d)*sq
rt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b*d^2))*d*abs(b)/b - 6*(sqrt(b^2*c + (b*x + a)*
b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^2*b*d^2)*log(a
bs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*c*abs(b)/b^2 - 6*(sqrt(b^2*
c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^
2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*d))*a*d*abs(b)/b^
3)/b

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b x} (c+d x)^{3/2} \, dx=\int \sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2} \,d x \]

[In]

int((a + b*x)^(1/2)*(c + d*x)^(3/2),x)

[Out]

int((a + b*x)^(1/2)*(c + d*x)^(3/2), x)